The phrase "finding the arithmetic progression of a number" is a bit ambiguous. It's more accurate to say you're finding or defining an arithmetic progression that includes a certain number. Here's how to approach finding or defining an arithmetic progression (AP), depending on what you're given:
Understanding Arithmetic Progression (AP)
An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). To define an AP, you generally need two pieces of information:
- First term (a): The initial number in the sequence.
- Common difference (d): The constant value added to each term to get the next.
Scenario 1: You are given the first term (a) and the common difference (d).
In this case, finding the arithmetic progression is straightforward. You simply start with the first term and repeatedly add the common difference to generate subsequent terms.
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Example:
- First term (a) = 2
- Common difference (d) = 3
The arithmetic progression is: 2, 5, 8, 11, 14, ...
Scenario 2: You are given a term (not necessarily the first) and the common difference (d).
If you know any term in the sequence (let's say the nth term, denoted as an) and the common difference (d), you can use the general formula to find other terms, including the first term.
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General formula for the nth term:
an = a + (n - 1)d
Where:
- an is the nth term
- a is the first term
- n is the position of the term in the sequence
- d is the common difference
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Example:
- a5 = 15 (the 5th term is 15)
- d = 2
To find the first term (a):
15 = a + (5 - 1)2
15 = a + 8
a = 7Now you know the first term (a = 7) and the common difference (d = 2), so the arithmetic progression is: 7, 9, 11, 13, 15, ...
Scenario 3: You are given two terms in the sequence.
Suppose you are given the mth term (am) and the nth term (an), where m ≠ n. You can find the common difference (d) using the following formula:
- d = (an - am) / (n - m)
Once you find the common difference, you can substitute it back into the general formula (an = a + (n - 1)d) along with either an or am to find the first term (a). Then, you can generate the AP as in Scenario 1.
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Example:
- a3 = 7 (the 3rd term is 7)
- a7 = 15 (the 7th term is 15)
To find the common difference (d):
d = (15 - 7) / (7 - 3)
d = 8 / 4
d = 2Now, find the first term (a) using a3:
7 = a + (3 - 1)2
7 = a + 4
a = 3The arithmetic progression is: 3, 5, 7, 9, 11, 13, 15, ...
Scenario 4: You only have one number.
If you are only given one number, there are infinite possible arithmetic progressions that include that number. You can choose any common difference you like, and then adjust the first term accordingly. For instance, if you're given the number 10, some possible arithmetic progressions are:
- d = 1: 4, 5, 6, 7, 8, 9, 10, 11, 12... (10 is the 7th term)
- d = 2: 2, 4, 6, 8, 10, 12, 14... (10 is the 5th term)
- d = -1: 16, 15, 14, 13, 12, 11, 10, 9, 8... (10 is the 7th term)
- d = 0: ...10, 10, 10, 10, 10... (10 is every term)
Without more information, you can't define a unique arithmetic progression.
In summary, finding an arithmetic progression requires knowing either the first term and common difference, two terms in the sequence, or one term and the common difference. If you only have one number, you can create infinitely many different arithmetic progressions containing it by varying the common difference.
